Integrals

Let f : R R be a function defined by f(x)= 4x4x+2 and M= f(a)f(1-a)x 4(x(1-x))dx, N= f(a)f(1-a) 4(x(1-x))dx; a 12. If M= N, , N, then the least value of 2+ 2 is equal to .

Let f, g:(0, ) R be two functions defined by f(x)= -xx (|t|-t2 )e-t2dt and g(x)= 0x2t1/2e-tdt. Then, the value of 9 (f ( e9 )+g ( e9 ) ) is equal to :

1. 10
2. 9
3. 8
4. 6

| 120 3 0 x2 x x 4x+ 4xdx | is equal to .

The value of n k=1n n3 (n2+k2 ) (n2+3k2 ) is :

1. 8(2 3+3)
2. (2 3+3) 24
3. 13 8(4 3+3)
4. 13(2 3-3) 8

The value of 9 09 [ 10xx+1 ]dx, where [t] denotes the greatest integer less than or equal to t, is .

Let f: R R be a function defined by f(x)= x (1+x4 )1/4, and g(x)=f(f(f(f(x)))). Then, 18 0 2 5x2g(x) dx is equal to

1. 36
2. 33
3. 39
4. 42

Let y=f(x) be a thrice differentiable function in (-5,5). Let the tangents to the curve y=f(x) at (1,f(1)) and (3,f(3)) make angles /6 and /4, respectively with positive x-axis. If 27 13 ( (f'(t) )2+1 )f''(t) dt= + 3 where , are integers, then the value of + equals

1. 26
2. -16
3. 36
4. -14

Let f: R R be defined as f(x)=ae2x+bex+cx. If f(0)=-1, f' ( e2 )=21 and 0 e4(f(x)-cx) dx= 392, then the value of |a+b+c| equals

1. 16
2. 12
3. 8
4. 10

For x (- 2, 2 ), if y(x)= cosec x+ xcosec x x+ x 2x dx, and x ( 2 )-y(x)=0 then y ( 4 ) is equal to

1. - 1 2 -1 ( 1 2 )
2. -1 ( 1 2 )
3. 12 -1 ( 1 2 )
4. 1 2 -1 (- 12 )

If the value of the integral - 2 2 ( x2 x1+ x+ 1+ 2x1+e x2023 )dx= 4( +a)-2, then the value of a is

1. - 32
2. 3
3. 32
4. 2